\(\int \frac {x^5}{(c+d x)^{3/2} (a-b x^2)^3} \, dx\) [725]

Optimal result

Integrand size = 23, antiderivative size = 350 \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=-\frac {2 c^5}{\left (b c^2-a d^2\right )^3 \sqrt {c+d x}}+\frac {a^2 \sqrt {c+d x} \left (a d^2+b c (c-2 d x)\right )}{4 b^2 \left (b c^2-a d^2\right )^2 \left (a-b x^2\right )^2}+\frac {a \sqrt {c+d x} \left (9 a^2 d^4-b^2 c^3 (16 c-35 d x)-7 a b c d^2 (3 c+d x)\right )}{16 b^2 \left (b c^2-a d^2\right )^3 \left (a-b x^2\right )}+\frac {\left (32 b c^2-22 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 b^{9/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{7/2}}+\frac {\left (32 b c^2+22 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 b^{9/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{7/2}} \] Output:

-2*c^5/(-a*d^2+b*c^2)^3/(d*x+c)^(1/2)+1/4*a^2*(d*x+c)^(1/2)*(a*d^2+b*c*(-2 
*d*x+c))/b^2/(-a*d^2+b*c^2)^2/(-b*x^2+a)^2+1/16*a*(d*x+c)^(1/2)*(9*a^2*d^4 
-b^2*c^3*(-35*d*x+16*c)-7*a*b*c*d^2*(d*x+3*c))/b^2/(-a*d^2+b*c^2)^3/(-b*x^ 
2+a)+1/32*(32*b*c^2-22*a^(1/2)*b^(1/2)*c*d+5*a*d^2)*arctanh(b^(1/4)*(d*x+c 
)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/b^(9/4)/(b^(1/2)*c-a^(1/2)*d)^(7/2)+1 
/32*(32*b*c^2+22*a^(1/2)*b^(1/2)*c*d+5*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2 
)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/b^(9/4)/(b^(1/2)*c+a^(1/2)*d)^(7/2)
 

Mathematica [A] (verified)

Time = 3.47 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.20 \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=\frac {\frac {2 \left (-32 b^4 c^5 x^4+5 a^4 d^4 (c+d x)+a b^3 c^3 x^2 \left (80 c^2-19 c d x-35 d^2 x^2\right )-a^3 b d^2 \left (21 c^3+20 c^2 d x+8 c d^2 x^2+9 d^3 x^3\right )+a^2 b^2 c \left (-44 c^4+15 c^3 d x+48 c^2 d^2 x^2+28 c d^3 x^3+7 d^4 x^4\right )\right )}{\left (b c^2-a d^2\right )^3 \sqrt {c+d x} \left (a-b x^2\right )^2}+\frac {\left (32 b c^2+22 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^3 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {\left (32 b c^2-22 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^3 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{32 b^2} \] Input:

Integrate[x^5/((c + d*x)^(3/2)*(a - b*x^2)^3),x]
 

Output:

((2*(-32*b^4*c^5*x^4 + 5*a^4*d^4*(c + d*x) + a*b^3*c^3*x^2*(80*c^2 - 19*c* 
d*x - 35*d^2*x^2) - a^3*b*d^2*(21*c^3 + 20*c^2*d*x + 8*c*d^2*x^2 + 9*d^3*x 
^3) + a^2*b^2*c*(-44*c^4 + 15*c^3*d*x + 48*c^2*d^2*x^2 + 28*c*d^3*x^3 + 7* 
d^4*x^4)))/((b*c^2 - a*d^2)^3*Sqrt[c + d*x]*(a - b*x^2)^2) + ((32*b*c^2 + 
22*Sqrt[a]*Sqrt[b]*c*d + 5*a*d^2)*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d] 
*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/((Sqrt[b]*c + Sqrt[a]*d)^3*Sqrt[ 
-(b*c) - Sqrt[a]*Sqrt[b]*d]) + ((32*b*c^2 - 22*Sqrt[a]*Sqrt[b]*c*d + 5*a*d 
^2)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - S 
qrt[a]*d)])/((Sqrt[b]*c - Sqrt[a]*d)^3*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]))/ 
(32*b^2)
 

Rubi [A] (verified)

Time = 3.86 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.56, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {561, 25, 27, 1673, 27, 2198, 27, 2195, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^5}{\left (a-b x^2\right )^3 (c+d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 561

\(\displaystyle \frac {2 \int \frac {x^5}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \int -\frac {x^5}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \int -\frac {d^5 x^5}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{d^6}\)

\(\Big \downarrow \) 1673

\(\displaystyle -\frac {2 \left (\frac {d^4 \int -\frac {2 \left (\frac {8 a b c^5}{d^2}+\frac {2 a \left (12 b^2 c^4-24 a b d^2 c^2+7 a^2 d^4\right ) (c+d x)^2 c}{d^2 \left (b c^2-a d^2\right )}-\frac {8 a \left (b c^2-a d^2\right ) (c+d x)^3}{d^2}-\frac {a \left (24 b^3 c^6-40 a b^2 d^2 c^4+3 a^2 b d^4 c^2+a^3 d^6\right ) (c+d x)}{b d^2 \left (b c^2-a d^2\right )}\right )}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (a d^2+3 b c^2-2 b c (c+d x)\right )}{8 b^2 \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^4 \int \frac {\frac {8 a b c^5}{d^2}+\frac {2 a \left (12 b^2 c^4-24 a b d^2 c^2+7 a^2 d^4\right ) (c+d x)^2 c}{d^2 \left (b c^2-a d^2\right )}-\frac {8 a \left (b c^2-a d^2\right ) (c+d x)^3}{d^2}-\frac {a \left (24 b^3 c^6-40 a b^2 d^2 c^4+3 a^2 b d^4 c^2+a^3 d^6\right ) (c+d x)}{b d^2 \left (b c^2-a d^2\right )}}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (a d^2+3 b c^2-2 b c (c+d x)\right )}{8 b^2 \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 2198

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (\frac {d^4 \int -\frac {2 \left (\frac {32 a^2 b^2 c^5}{d^4}-\frac {7 a^3 b \left (5 b c^2-a d^2\right ) (c+d x)^2 c}{d^2 \left (b c^2-a d^2\right )}-\frac {a^2 \left (32 b^3 c^6-109 a b^2 d^2 c^4+26 a^2 b d^4 c^2-5 a^3 d^6\right ) (c+d x)}{d^4 \left (b c^2-a d^2\right )}\right )}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^2 \sqrt {c+d x} \left (-9 a^2 d^4-7 b c (c+d x) \left (5 b c^2-a d^2\right )+14 a b c^2 d^2+51 b^2 c^4\right )}{4 b \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (a d^2+3 b c^2-2 b c (c+d x)\right )}{8 b^2 \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {d^4 \int \frac {\frac {32 a^2 b^2 c^5}{d^4}-\frac {7 a^3 b \left (5 b c^2-a d^2\right ) (c+d x)^2 c}{d^2 \left (b c^2-a d^2\right )}-\frac {a^2 \left (32 b^3 c^6-109 a b^2 d^2 c^4+26 a^2 b d^4 c^2-5 a^3 d^6\right ) (c+d x)}{d^4 \left (b c^2-a d^2\right )}}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^2 \sqrt {c+d x} \left (-9 a^2 d^4-7 b c (c+d x) \left (5 b c^2-a d^2\right )+14 a b c^2 d^2+51 b^2 c^4\right )}{4 b \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (a d^2+3 b c^2-2 b c (c+d x)\right )}{8 b^2 \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 2195

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {d^4 \int \left (\frac {32 a^2 b^2 c^5}{d^2 \left (a d^2-b c^2\right ) (c+d x)}+\frac {a^2 \left (-32 b^3 c^6-109 a b^2 d^2 c^4+26 a^2 b d^4 c^2+b \left (32 b^2 c^4+35 a b d^2 c^2-7 a^2 d^4\right ) (c+d x) c-5 a^3 d^6\right )}{d^2 \left (b c^2-a d^2\right ) \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}\right )d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^2 \sqrt {c+d x} \left (-9 a^2 d^4-7 b c (c+d x) \left (5 b c^2-a d^2\right )+14 a b c^2 d^2+51 b^2 c^4\right )}{4 b \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (a d^2+3 b c^2-2 b c (c+d x)\right )}{8 b^2 \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {d^4 \left (-\frac {a^2 \left (\sqrt {a} d+\sqrt {b} c\right )^2 \left (-22 \sqrt {a} \sqrt {b} c d+5 a d^2+32 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt [4]{b} d^2 \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}-\frac {a^2 \left (\sqrt {b} c-\sqrt {a} d\right )^2 \left (22 \sqrt {a} \sqrt {b} c d+5 a d^2+32 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt [4]{b} d^2 \left (\sqrt {a} d+\sqrt {b} c\right )^{3/2}}+\frac {32 a^2 b^2 c^5}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{4 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^2 \sqrt {c+d x} \left (-9 a^2 d^4-7 b c (c+d x) \left (5 b c^2-a d^2\right )+14 a b c^2 d^2+51 b^2 c^4\right )}{4 b \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a^2 d^6 \sqrt {c+d x} \left (a d^2+3 b c^2-2 b c (c+d x)\right )}{8 b^2 \left (b c^2-a d^2\right )^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^6}\)

Input:

Int[x^5/((c + d*x)^(3/2)*(a - b*x^2)^3),x]
 

Output:

(-2*(-1/8*(a^2*d^6*Sqrt[c + d*x]*(3*b*c^2 + a*d^2 - 2*b*c*(c + d*x)))/(b^2 
*(b*c^2 - a*d^2)^2*(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x) 
^2)/d^2)^2) - (d^4*(-1/4*(a^2*d^2*Sqrt[c + d*x]*(51*b^2*c^4 + 14*a*b*c^2*d 
^2 - 9*a^2*d^4 - 7*b*c*(5*b*c^2 - a*d^2)*(c + d*x)))/(b*(b*c^2 - a*d^2)^2* 
(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)) - (d^4*(( 
32*a^2*b^2*c^5)/(d^2*(b*c^2 - a*d^2)*Sqrt[c + d*x]) - (a^2*(Sqrt[b]*c + Sq 
rt[a]*d)^2*(32*b*c^2 - 22*Sqrt[a]*Sqrt[b]*c*d + 5*a*d^2)*ArcTanh[(b^(1/4)* 
Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*b^(1/4)*d^2*(Sqrt[b]*c - S 
qrt[a]*d)^(3/2)) - (a^2*(Sqrt[b]*c - Sqrt[a]*d)^2*(32*b*c^2 + 22*Sqrt[a]*S 
qrt[b]*c*d + 5*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqr 
t[a]*d]])/(2*b^(1/4)*d^2*(Sqrt[b]*c + Sqrt[a]*d)^(3/2))))/(4*a*b*(b*c^2 - 
a*d^2))))/(8*a*b*(b*c^2 - a*d^2))))/d^6
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1673

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 
4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q 
, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^ 
2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a 
*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), 
 x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[x^m*(a + b*x^2 + c*x^4)^(p + 
 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + 
 e*x^2)^q, a + b*x^2 + c*x^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5 
) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x], x], x]] /; Fr 
eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] 
&& ILtQ[m/2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2195

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ 
Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; 
FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
 

rule 2198

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol 
ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial 
Remainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4) 
^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 
 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[x^m*(a + b*x^2 + 
 c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*Qx)/x^m + (b^2*d*(2* 
p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - 
m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x 
^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.43

method	result	size
pseudoelliptic	\(-\frac {5 \left (-\frac {\left (\left (-\frac {7}{5} a^{2} c \,d^{4}+7 a \,c^{3} d^{2} b +\frac {32}{5} c^{5} b^{2}\right ) \sqrt {a b \,d^{2}}+a \,d^{2} \left (a^{2} d^{4}-\frac {19}{5} b \,c^{2} d^{2} a +\frac {74}{5} b^{2} c^{4}\right )\right ) b \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2}+\left (-\frac {\left (\left (\frac {7}{5} a^{2} c \,d^{4}-7 a \,c^{3} d^{2} b -\frac {32}{5} c^{5} b^{2}\right ) \sqrt {a b \,d^{2}}+a \,d^{2} \left (a^{2} d^{4}-\frac {19}{5} b \,c^{2} d^{2} a +\frac {74}{5} b^{2} c^{4}\right )\right ) b \sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2}+\left (-\frac {32 b^{4} c^{5} x^{4}}{5}+16 x^{2} \left (-\frac {7}{16} d^{2} x^{2}-\frac {19}{80} c d x +c^{2}\right ) a \,c^{3} b^{3}-\frac {44 a^{2} c \left (-\frac {7}{44} d^{4} x^{4}-\frac {7}{11} c \,d^{3} x^{3}-\frac {12}{11} d^{2} c^{2} x^{2}-\frac {15}{44} c^{3} d x +c^{4}\right ) b^{2}}{5}-\frac {21 \left (d x +c \right ) d^{2} \left (\frac {3}{7} d^{2} x^{2}-\frac {1}{21} c d x +c^{2}\right ) a^{3} b}{5}+a^{4} d^{4} \left (d x +c \right )\right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\right )}{16 \sqrt {a b \,d^{2}}\, \sqrt {d x +c}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right )^{2} \left (a \,d^{2}-b \,c^{2}\right )^{3} b^{2}}\)	\(502\)
derivativedivides	\(\frac {\frac {2 \left (\left (-\frac {7}{32} a^{2} c \,d^{4}+\frac {35}{32} a \,c^{3} d^{2} b \right ) \left (d x +c \right )^{\frac {7}{2}}+\frac {a \,d^{2} \left (9 a^{2} d^{4}-121 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32 b}-\frac {a c \,d^{2} \left (19 a^{2} d^{4}+6 b \,c^{2} d^{2} a -137 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b}-\frac {a \,d^{2} \left (5 a^{3} d^{6}-31 a^{2} b \,c^{2} d^{4}-25 a \,b^{2} c^{4} d^{2}+51 b^{3} c^{6}\right ) \sqrt {d x +c}}{32 b^{2}}\right )}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {\frac {\left (5 a^{3} d^{6}-19 a^{2} b \,c^{2} d^{4}+74 a \,b^{2} c^{4} d^{2}-7 \sqrt {a b \,d^{2}}\, a^{2} c \,d^{4}+35 \sqrt {a b \,d^{2}}\, a b \,c^{3} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{5}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-5 a^{3} d^{6}+19 a^{2} b \,c^{2} d^{4}-74 a \,b^{2} c^{4} d^{2}-7 \sqrt {a b \,d^{2}}\, a^{2} c \,d^{4}+35 \sqrt {a b \,d^{2}}\, a b \,c^{3} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{5}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 b}}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}+\frac {2 c^{5}}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}\)	\(522\)
default	\(\frac {\frac {2 \left (\left (-\frac {7}{32} a^{2} c \,d^{4}+\frac {35}{32} a \,c^{3} d^{2} b \right ) \left (d x +c \right )^{\frac {7}{2}}+\frac {a \,d^{2} \left (9 a^{2} d^{4}-121 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {5}{2}}}{32 b}-\frac {a c \,d^{2} \left (19 a^{2} d^{4}+6 b \,c^{2} d^{2} a -137 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b}-\frac {a \,d^{2} \left (5 a^{3} d^{6}-31 a^{2} b \,c^{2} d^{4}-25 a \,b^{2} c^{4} d^{2}+51 b^{3} c^{6}\right ) \sqrt {d x +c}}{32 b^{2}}\right )}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {\frac {\left (5 a^{3} d^{6}-19 a^{2} b \,c^{2} d^{4}+74 a \,b^{2} c^{4} d^{2}-7 \sqrt {a b \,d^{2}}\, a^{2} c \,d^{4}+35 \sqrt {a b \,d^{2}}\, a b \,c^{3} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{5}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-5 a^{3} d^{6}+19 a^{2} b \,c^{2} d^{4}-74 a \,b^{2} c^{4} d^{2}-7 \sqrt {a b \,d^{2}}\, a^{2} c \,d^{4}+35 \sqrt {a b \,d^{2}}\, a b \,c^{3} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{5}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 b}}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}+\frac {2 c^{5}}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}\)	\(522\)

Input:

int(x^5/(d*x+c)^(3/2)/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-5/16*(-1/2*((-7/5*a^2*c*d^4+7*a*c^3*d^2*b+32/5*c^5*b^2)*(a*b*d^2)^(1/2)+a 
*d^2*(a^2*d^4-19/5*b*c^2*d^2*a+74/5*b^2*c^4))*b*((b*c+(a*b*d^2)^(1/2))*b)^ 
(1/2)*(d*x+c)^(1/2)*(-b*x^2+a)^2*arctan(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^( 
1/2))*b)^(1/2))+(-1/2*((7/5*a^2*c*d^4-7*a*c^3*d^2*b-32/5*c^5*b^2)*(a*b*d^2 
)^(1/2)+a*d^2*(a^2*d^4-19/5*b*c^2*d^2*a+74/5*b^2*c^4))*b*(d*x+c)^(1/2)*(-b 
*x^2+a)^2*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))+(-32/5* 
b^4*c^5*x^4+16*x^2*(-7/16*d^2*x^2-19/80*c*d*x+c^2)*a*c^3*b^3-44/5*a^2*c*(- 
7/44*d^4*x^4-7/11*c*d^3*x^3-12/11*d^2*c^2*x^2-15/44*c^3*d*x+c^4)*b^2-21/5* 
(d*x+c)*d^2*(3/7*d^2*x^2-1/21*c*d*x+c^2)*a^3*b+a^4*d^4*(d*x+c))*(a*b*d^2)^ 
(1/2)*((b*c+(a*b*d^2)^(1/2))*b)^(1/2))*((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))/( 
a*b*d^2)^(1/2)/(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)/((-b*c+(a*b*d 
^2)^(1/2))*b)^(1/2)/(-b*x^2+a)^2/(a*d^2-b*c^2)^3/b^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9592 vs. \(2 (296) = 592\).

Time = 26.55 (sec) , antiderivative size = 9592, normalized size of antiderivative = 27.41 \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(d*x+c)^(3/2)/(-b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**5/(d*x+c)**(3/2)/(-b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=\int { -\frac {x^{5}}{{\left (b x^{2} - a\right )}^{3} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^5/(d*x+c)^(3/2)/(-b*x^2+a)^3,x, algorithm="maxima")
 

Output:

-integrate(x^5/((b*x^2 - a)^3*(d*x + c)^(3/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2229 vs. \(2 (296) = 592\).

Time = 0.47 (sec) , antiderivative size = 2229, normalized size of antiderivative = 6.37 \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(d*x+c)^(3/2)/(-b*x^2+a)^3,x, algorithm="giac")
 

Output:

-2*c^5/((b^3*c^6 - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 - a^3*d^6)*sqrt(d*x + 
 c)) + 1/32*((b^5*c^6*d - 3*a*b^4*c^4*d^3 + 3*a^2*b^3*c^2*d^5 - a^3*b^2*d^ 
7)^2*(32*sqrt(a*b)*b^2*c^5 + 35*sqrt(a*b)*a*b*c^3*d^2 - 7*sqrt(a*b)*a^2*c* 
d^4)*abs(b) - (32*b^8*c^12 + 13*a*b^7*c^10*d^2 - 257*a^2*b^6*c^8*d^4 + 378 
*a^3*b^5*c^6*d^6 - 202*a^4*b^4*c^4*d^8 + 41*a^5*b^3*c^2*d^10 - 5*a^6*b^2*d 
^12)*abs(b^5*c^6*d - 3*a*b^4*c^4*d^3 + 3*a^2*b^3*c^2*d^5 - a^3*b^2*d^7)*ab 
s(b) + (74*sqrt(a*b)*b^12*c^17*d^2 - 463*sqrt(a*b)*a*b^11*c^15*d^4 + 1229* 
sqrt(a*b)*a^2*b^10*c^13*d^6 - 1795*sqrt(a*b)*a^3*b^9*c^11*d^8 + 1565*sqrt( 
a*b)*a^4*b^8*c^9*d^10 - 829*sqrt(a*b)*a^5*b^7*c^7*d^12 + 263*sqrt(a*b)*a^6 
*b^6*c^5*d^14 - 49*sqrt(a*b)*a^7*b^5*c^3*d^16 + 5*sqrt(a*b)*a^8*b^4*c*d^18 
)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^6*c^7 - 3*a*b^5*c^5*d^2 + 3*a^2*b^ 
4*c^3*d^4 - a^3*b^3*c*d^6 + sqrt((b^6*c^7 - 3*a*b^5*c^5*d^2 + 3*a^2*b^4*c^ 
3*d^4 - a^3*b^3*c*d^6)^2 - (b^6*c^8 - 4*a*b^5*c^6*d^2 + 6*a^2*b^4*c^4*d^4 
- 4*a^3*b^3*c^2*d^6 + a^4*b^2*d^8)*(b^6*c^6 - 3*a*b^5*c^4*d^2 + 3*a^2*b^4* 
c^2*d^4 - a^3*b^3*d^6)))/(b^6*c^6 - 3*a*b^5*c^4*d^2 + 3*a^2*b^4*c^2*d^4 - 
a^3*b^3*d^6)))/((b^11*c^13 - 6*a*b^10*c^11*d^2 + 15*a^2*b^9*c^9*d^4 - 20*a 
^3*b^8*c^7*d^6 + 15*a^4*b^7*c^5*d^8 - 6*a^5*b^6*c^3*d^10 + a^6*b^5*c*d^12 
- sqrt(a*b)*b^10*c^12*d + 6*sqrt(a*b)*a*b^9*c^10*d^3 - 15*sqrt(a*b)*a^2*b^ 
8*c^8*d^5 + 20*sqrt(a*b)*a^3*b^7*c^6*d^7 - 15*sqrt(a*b)*a^4*b^6*c^4*d^9 + 
6*sqrt(a*b)*a^5*b^5*c^2*d^11 - sqrt(a*b)*a^6*b^4*d^13)*sqrt(-b^2*c - sq...
 

Mupad [B] (verification not implemented)

Time = 14.50 (sec) , antiderivative size = 13735, normalized size of antiderivative = 39.24 \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(x^5/((a - b*x^2)^3*(c + d*x)^(3/2)),x)
 

Output:

((2*c^5)/(a*d^2 - b*c^2) + ((c + d*x)^4*(32*b^2*c^5 - 7*a^2*c*d^4 + 35*a*b 
*c^3*d^2))/(16*(a*d^2 - b*c^2)^3) + ((c + d*x)*(128*b^3*c^6 - 5*a^3*d^6 + 
51*a*b^2*c^4*d^2 + 26*a^2*b*c^2*d^4))/(16*b^2*(a*d^2 - b*c^2)^2) - ((c + d 
*x)^3*(128*b^3*c^6 - 9*a^3*d^6 + 121*a*b^2*c^4*d^2))/(16*b*(a*d^2 - b*c^2) 
^3) - (c*(c + d*x)^2*(19*a^3*d^6 - 192*b^3*c^6 - 73*a*b^2*c^4*d^2 + 6*a^2* 
b*c^2*d^4))/(16*b*(a*d^2 - b*c^2)^3))/(b^2*(c + d*x)^(9/2) - (4*b^2*c^3 - 
4*a*b*c*d^2)*(c + d*x)^(3/2) + (c + d*x)^(1/2)*(a^2*d^4 + b^2*c^4 - 2*a*b* 
c^2*d^2) + (6*b^2*c^2 - 2*a*b*d^2)*(c + d*x)^(5/2) - 4*b^2*c*(c + d*x)^(7/ 
2)) - atan(((((1024*b^10*c^11 - 25*a^5*d^11*(a*b^9)^(1/2) + 12452*a*b^9*c^ 
9*d^2 - 45*a^5*b^5*c*d^10 - 5760*b^5*c^10*d*(a*b^9)^(1/2) + 1929*a^2*b^8*c 
^7*d^4 - 1435*a^3*b^7*c^5*d^6 + 475*a^4*b^6*c^3*d^8 + 4081*a^2*b^3*c^6*d^5 
*(a*b^9)^(1/2) - 1227*a^3*b^2*c^4*d^7*(a*b^9)^(1/2) - 11680*a*b^4*c^8*d^3* 
(a*b^9)^(1/2) + 211*a^4*b*c^2*d^9*(a*b^9)^(1/2))/(4096*(b^16*c^14 - a^7*b^ 
9*d^14 - 7*a*b^15*c^12*d^2 + 21*a^2*b^14*c^10*d^4 - 35*a^3*b^13*c^8*d^6 + 
35*a^4*b^12*c^6*d^8 - 21*a^5*b^11*c^4*d^10 + 7*a^6*b^10*c^2*d^12)))^(1/2)* 
(5242880*a^16*b^11*d^32 + (c + d*x)^(1/2)*((1024*b^10*c^11 - 25*a^5*d^11*( 
a*b^9)^(1/2) + 12452*a*b^9*c^9*d^2 - 45*a^5*b^5*c*d^10 - 5760*b^5*c^10*d*( 
a*b^9)^(1/2) + 1929*a^2*b^8*c^7*d^4 - 1435*a^3*b^7*c^5*d^6 + 475*a^4*b^6*c 
^3*d^8 + 4081*a^2*b^3*c^6*d^5*(a*b^9)^(1/2) - 1227*a^3*b^2*c^4*d^7*(a*b^9) 
^(1/2) - 11680*a*b^4*c^8*d^3*(a*b^9)^(1/2) + 211*a^4*b*c^2*d^9*(a*b^9)^...
 

Reduce [B] (verification not implemented)

Time = 3.22 (sec) , antiderivative size = 4103, normalized size of antiderivative = 11.72 \[ \int \frac {x^5}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^5/(d*x+c)^(3/2)/(-b*x^2+a)^3,x)
 

Output:

( - 4*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d 
*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**4*b*c*d**5 + 32*sqrt(a) 
*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt( 
b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*b**2*c**3*d**3 + 8*sqrt(a)*sqrt(c 
+ d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt( 
sqrt(b)*sqrt(a)*d - b*c)))*a**3*b**2*c*d**5*x**2 + 212*sqrt(a)*sqrt(c + d* 
x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt 
(b)*sqrt(a)*d - b*c)))*a**2*b**3*c**5*d - 64*sqrt(a)*sqrt(c + d*x)*sqrt(sq 
rt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a 
)*d - b*c)))*a**2*b**3*c**3*d**3*x**2 - 4*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt( 
b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d 
 - b*c)))*a**2*b**3*c*d**5*x**4 - 424*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*s 
qrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b 
*c)))*a*b**4*c**5*d*x**2 + 32*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d 
 - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a* 
b**4*c**3*d**3*x**4 + 212*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b 
*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b**5*c 
**5*d*x**4 + 10*sqrt(b)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan(( 
sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**5*d**6 - 52*s 
qrt(b)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*...